Refresh just pulls in another copy of the data at the URL given – it will not change the URL
You would need a redirect so requests to the old URL are sent to the new one – normally done through the web server, and definitely not the browser

Sounds like you want to set up web redirection? You can usually do this by logging into the registrar that hosts the old site and using their forwarding tools to point the old site to the IP address of the new site.